Physical chemistry

Transition state theory

Reaction rates as a passage over a saddle point: the activated complex, the Eyring equation, activation thermodynamics (ΔG‡, ΔH‡, ΔS‡), and where the theory fails — recrossing, tunnelling and dynamics.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: a mountain pass between two valleys

Reactants and products are two valleys; the route between them crosses a pass. Molecules do not stroll across the landscape at random — they pour along the cheapest crossing, the lowest pass, called the transition state or activated complex. The rate of reaction is then a traffic problem: how much molecular population reaches the pass per second, weighted by Boltzmann statistics. Catalysts are not magic: they reroute the path through a lower pass.

Rotate the surface: the transition state is a maximum along the dashed minimum-energy path but a minimum across it — the signature of a saddle.

SchoolSchool: from Arrhenius to the barrier as a molecular object

Arrhenius gave a recipe — rate ∝ e^(−Ea/RT) — without saying what the barrier is made of. Transition state theory (Eyring, Evans–Polanyi, 1935) answers: the barrier top is a real configuration of nuclei, the activated complex, and the rate constant follows from counting how many complexes sit at the col and how fast they cross. The central assumptions: molecules reaching the col are in quasi-equilibrium with reactants, and once through the col they do not turn back.

Definition: Activated complex and dividing surface

The activated complex ‡ is the molecular configuration at the top of the minimum-energy path — a maximum along the reaction coordinate, a minimum in every orthogonal direction (one imaginary vibrational frequency, all others real). In modern language it is a dividing surface in phase space: trajectories crossing it from reactants are counted as reactive events. It is not an isolable species — its "lifetime" is the duration of one crossing, ~10⁻¹³ s.

Example: Reading an Eyring activation parameter

A bimolecular association A + B → AB in solution has ΔH‡ = 45 kJ mol⁻¹ and ΔS‡ = −80 J mol⁻¹ K⁻¹. What does the large negative entropy of activation say, and what is ΔG‡ at 298 K?

Solution

ΔG‡ = ΔH‡ − TΔS‡ = 45 − 298×(−0.080) = 45 + 23.8 ≈ 69 kJ mol⁻¹. The strongly negative ΔS‡ means two partners lose translational and rotational freedom when they lock into one complex — a tight, ordered transition state. Associations, cyclisations and concerted pericyclic reactions typically have ΔS‡ < 0; dissociations that loosen structure have ΔS‡ > 0. The entropy term here adds ~24 kJ mol⁻¹ to the barrier at 298 K.

UndergraduateUniversity: the Eyring equation

The derivation is short and beautiful. Assume the activated complex is in equilibrium with reactants, governed by an equilibrium constant K‡ built from partition functions; assume complexes cross the col at frequency ν, with the loose vibration along the reaction coordinate becoming the crossing motion. The equilibrium population at the col times the crossing frequency gives the rate. The result packages the barrier as a free energy:

k=\kappa\frac{k_BT}{h}\frac{Q^{\ddagger}}{Q_{A}Q_{B}}\,e^{-\varepsilon_0/k_BT}=\kappa\frac{k_BT}{h}K^{\ddagger}=\kappa\frac{k_BT}{h}e^{-\Delta G^{\ddagger}/RT}

\ln\frac{k}{T}=\ln\frac{k_B}{h}+\frac{\Delta S^{\ddagger}}{R}-\frac{\Delta H^{\ddagger}}{R}\frac{1}{T}\qquad(\text{Eyring plot})

The prefactor k_BT/h ≈ 6.2×10¹² s⁻¹ at 298 K is the universal attempt frequency — every activated complex tries to cross about this often. ΔH‡ relates to the Arrhenius activation energy through Ea = ΔH‡ + RT (for a unimolecular gas reaction), and ΔS‡ reshapes the pre-exponential factor: this is how TST explains why some reactions are "slow for their barrier" — an ordered, tight transition state costs entropy before energy.

Arrhenius vs Eyring vocabulary
Arrhenius	Eyring / TST
Ea (from slope of ln k vs 1/T)	ΔH‡ = Ea − RT (+RT per molecularity)
Pre-exponential factor A	(k_BT/h)·e^(ΔS‡/R) — contains entropy
Empirical parameters	ΔG‡, ΔH‡, ΔS‡ — thermodynamic content

Vary ΔH‡ (barrier height) or ΔS‡ (ordering of the transition state) and watch how the rate constant responds over 200–800 K.

AdvancedAdvanced: what the theory ignores — recrossing, tunnelling, dynamics

TST's no-return assumption overestimates rates: real trajectories can cross the col and be knocked back (recrossing), so the true rate is κ·k_TST with transmission coefficient κ < 1. Kramers theory adds solvent friction; variational TST moves the dividing surface along the reaction coordinate to minimise the flux; in enzymes and condensed phases, surface diffusion and cage dynamics dominate. Conversely, light atoms cheat the barrier entirely: hydrogen tunnels through it.

k_{\mathrm{TST}}^{\mathrm{tun}}\approx k_{\mathrm{TST}}\,\kappa_{\mathrm{W}},\quad \kappa_{\mathrm{W}}=1+\frac{1}{24}\left(\frac{h\nu^{\ddagger}}{k_BT}\right)^{\!2}\quad(\text{Wigner})

Kinetic isotope effects expose tunnelling directly: replacing H with D makes C–D bonds ~7-fold slower to break near room temperature classically, but tunnelling can amplify k_H/k_D to 50 or more and give anomalous Arrhenius slopes. RRKM theory extends TST inside the molecule: for unimolecular reactions the rate depends on how energy sloshes among internal modes before the critical bond dissociates — microcanonical TST where the bottleneck is internal energy redistribution, not a single surface crossing.

ResearchResearch frontier

References

The Activated Complex in Chemical Reactions · H. Eyring, 1935
Some Applications of the Transition State Method to the Calculation of Reaction Velocities, Especially in Solution · M. G. Evans, M. Polanyi, 1935
Roaming atoms and radicals: a new mechanism in molecular dissociation · D. Townsend, S. A. Lahankar, S. K. Lee, S. D. Chambreau, A. G. Suits, X. Zhang, J. Rheinecker, L. B. Harding, J. M. Bowman, 2004
Reaction-rate theory: fifty years after Kramers · P. Hänggi, P. Talkner, M. Borkovec, 1990